Summary of Research Accomplishments and Goals

My Ph.D. thesis centered on a class of univariate polynomials over finite fields known as composed products or root-based compositions . Work related to the thesis appears in [5], [6], and [41], while additional results pertaining to multivariate analogues appear in [48] and [49]. My post-thesis research in finite fields includes not only work with polynomials (see [14] and [44]), but also work involving more general functions [23], sequences [25], primitive elements possessing certain properties [40], and exponential sums [42]. Moreover, I have expanded my research horizons in other areas of algebra and discrete mathematics, particularly in the area of combinatorial number theory. I have written papers dealing with integer sequences (see [43], [45] and [46]), and am currently writing a paper dealing with polynomial sequences formed by Fibonacci or Lucas numbers, with the goal of extending this work to polynomial sequences formed by Lucas sequence pairs. The purpose of this document is to give a brief introduction to each subject, list my contributions in each area, and present objectives for future work.